One of two ways...and the individual States would decide.
1. For every private sale...the parties go to a dealer. This method, I am not a big fan of. I can see huge burdens in our more sparsely inhabited States.
2. Make it part of the Driver's license/Photo ID process. You get your.license renewed, the background check is done...if you pass, you get a checkmark under "firearms eligible" or what have you. If you are convicted of a violent crime or deemed insane, that checkmark is removed from your license...the check would be readministered every time you renew. This makes the most sense to me.